{"id":1482,"date":"2023-06-22T02:36:51","date_gmt":"2023-06-22T02:36:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/multiple-linear-regression-learn-it-2\/"},"modified":"2025-02-19T21:39:39","modified_gmt":"2025-02-19T21:39:39","slug":"multiple-linear-regression-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/multiple-linear-regression-learn-it-2\/","title":{"raw":"Multiple Linear Regression - Learn It 2","rendered":"Multiple Linear Regression &#8211; Learn It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write and describe a multiple linear regression model equation&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Write and describe a multiple linear regression model equation<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate and describe the unadjusted coefficient of determination&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate and describe the unadjusted coefficient of determination<\/span><\/li>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Assess the model assumptions with a residual or a predicted values plot&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Assess the model assumptions with a residual or a predicted values plot<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<section>Previously, we have learned about linear regression to describe the relationship between two quantitative variables:\r\n\r\n<div>\r\n<ul>\r\n\t<li>An\u00a0explanatory variable\u00a0is an\u00a0independent variable, one that may explain or cause a change in another variable.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n<ul>\r\n\t<li>A\u00a0response variable\u00a0is a\u00a0dependent variable, one that changes in response to the explanatory variable.<\/li>\r\n<\/ul>\r\n<p>However, what if we found that there are more than one explanatory variable that explain the response variable?<\/p>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>multiple linear regression model<\/h3>\r\n<p>A linear regression model with two or more explanatory variables is called a <strong>multiple linear regression model. <\/strong>Since there is more than one explanatory variable, the model is no longer a line. In fact, we can include [latex]p[\/latex]\u00a0explanatory variables in our model.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The equation for the estimated model that uses [latex]p[\/latex]\u00a0variables is<\/p>\r\n<p style=\"text-align: center;\">[latex]\\hat{y} = a + b_1 \\cdot x_1 + b_2 \\cdot x_2 + ... + b_p \\cdot x_p[\/latex]<\/p>\r\n<p>where [latex]b_1, b_2, ... ,b_p[\/latex] are the regression coefficients for explanatory variables [latex]x_1, x_2, ... ,x_p[\/latex], respectively.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>In multiple linear regression, [latex]b_1, b_2, ... , b_p[\/latex] are called <strong>partial slopes<\/strong>.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3232[\/ohm2_question]<\/section>\r\n<p>We can interpret the regression coefficients for each explanatory variable in the model in terms of the relationship with the response variable. The explanation is very similar to what we have seen in simple linear regression models. However, since it is a partial slope, we have to make sure that we hold any other explanatory variables constant in our interpretation.<\/p>\r\n<section class=\"textbox proTip\">For the following regression equation,\r\n\r\n<p style=\"text-align: center;\">[latex]\\hat{y} = a + b_1 \\cdot x_1 + b_2 \\cdot x_2 + ... + b_p \\cdot x_p[\/latex]<\/p>\r\n<p>the partial slope, [latex]b_1[\/latex], represents the expected change in the response variable, [latex]y[\/latex], for every one unit increase in [latex]x_1[\/latex], holding explanatory variables [latex]x_1, x_2, ... , x_p[\/latex] constant.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3233[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Write and describe a multiple linear regression model equation&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Write and describe a multiple linear regression model equation<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate and describe the unadjusted coefficient of determination&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Calculate and describe the unadjusted coefficient of determination<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Assess the model assumptions with a residual or a predicted values plot&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;}\">Assess the model assumptions with a residual or a predicted values plot<\/span><\/li>\n<\/ul>\n<\/section>\n<section>Previously, we have learned about linear regression to describe the relationship between two quantitative variables:<\/p>\n<div>\n<ul>\n<li>An\u00a0explanatory variable\u00a0is an\u00a0independent variable, one that may explain or cause a change in another variable.<\/li>\n<\/ul>\n<\/div>\n<div>\n<ul>\n<li>A\u00a0response variable\u00a0is a\u00a0dependent variable, one that changes in response to the explanatory variable.<\/li>\n<\/ul>\n<p>However, what if we found that there are more than one explanatory variable that explain the response variable?<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>multiple linear regression model<\/h3>\n<p>A linear regression model with two or more explanatory variables is called a <strong>multiple linear regression model. <\/strong>Since there is more than one explanatory variable, the model is no longer a line. In fact, we can include [latex]p[\/latex]\u00a0explanatory variables in our model.<\/p>\n<p>&nbsp;<\/p>\n<p>The equation for the estimated model that uses [latex]p[\/latex]\u00a0variables is<\/p>\n<p style=\"text-align: center;\">[latex]\\hat{y} = a + b_1 \\cdot x_1 + b_2 \\cdot x_2 + ... + b_p \\cdot x_p[\/latex]<\/p>\n<p>where [latex]b_1, b_2, ... ,b_p[\/latex] are the regression coefficients for explanatory variables [latex]x_1, x_2, ... ,x_p[\/latex], respectively.<\/p>\n<p>&nbsp;<\/p>\n<p>In multiple linear regression, [latex]b_1, b_2, ... , b_p[\/latex] are called <strong>partial slopes<\/strong>.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3232\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3232&theme=lumen&iframe_resize_id=ohm3232&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>We can interpret the regression coefficients for each explanatory variable in the model in terms of the relationship with the response variable. The explanation is very similar to what we have seen in simple linear regression models. However, since it is a partial slope, we have to make sure that we hold any other explanatory variables constant in our interpretation.<\/p>\n<section class=\"textbox proTip\">For the following regression equation,<\/p>\n<p style=\"text-align: center;\">[latex]\\hat{y} = a + b_1 \\cdot x_1 + b_2 \\cdot x_2 + ... + b_p \\cdot x_p[\/latex]<\/p>\n<p>the partial slope, [latex]b_1[\/latex], represents the expected change in the response variable, [latex]y[\/latex], for every one unit increase in [latex]x_1[\/latex], holding explanatory variables [latex]x_1, x_2, ... , x_p[\/latex] constant.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3233\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3233&theme=lumen&iframe_resize_id=ohm3233&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":8,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":1473,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1482"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1482\/revisions"}],"predecessor-version":[{"id":6292,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1482\/revisions\/6292"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/1473"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1482\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1482"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1482"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1482"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1482"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}